1. Field of Invention
This invention relates to optical signal processing and, more particularly, to the use of fractional Fourier transform properties of lenses with traditional non-phase-shifting optical elements within traditional Fourier optical signal processing environments to realize, or closely approximate, arbitrary non-positive-definite transfer functions. The system and method herein can be applied to conventional lens-based optical image processing systems as well as to systems with other types of elements obeying Fractional Fourier optical models and as well to widely ranging environments such as integrated optics, optical computing systems, particle beam systems, radiation accelerators, and astronomical observation methods.
2. Background of the Invention
A number of references are cited herein; these are provided in a numbered list at the end of the Detailed Description. These references are cited as needed through the text by reference number(s) enclosed in square brackets. Further, the cited disclosure contained within reference [1-19] is hereby incorporated by reference.
The Fourier transforming properties of simple lenses and related optical elements is well known and heavily used in a branch of engineering known as xe2x80x9cFourier Opticsxe2x80x9d [1,2]. Classical Fourier Optics [1,2,3,4] allows for flexible signal processing of images by (1) using lenses or other means to take a first two-dimensional Fourier transform of an optical wavefront, thus creating at a particular spatial location a xe2x80x9cFourier planexe2x80x9d wherein the amplitude distribution of an original two-dimensional optical image becomes the two-dimensional Fourier transform of itself, (2) using a translucent plate or similar means in this location to introduce an optical transfer function operation on the optical wavefront, and (3) using lenses or other means to take a second Fourier transform which, within possible scaling and orientation differences, amounts to the convolution of the impulse response corresponding to the optical transfer function with the original image. In this way images can be relatively easily and inexpensively lowpass-filtered (details softened) and highpass-filtered (details enhanced) as well as multitude of other possibilities. These multitudes of possibilities have, due to properties of materials and fabrication limitations in transcending them, been limited to transfer functions that mathematically are xe2x80x9cpositive-definite,xe2x80x9d i.e. those which affect only amplitude and do not introduce varying phase relationships.
The Fractional Fourier transform has been independently developed several times over the years [5,7,8,9,10,14,15] and is related to several other mathematical objects such as the Bargmann transform [8] and the Hermite semigroup [13]. As shown in [5], the most general form of optical properties of lenses and other related elements [1,2,3] can be transformed into a Fractional Fourier transform representation. This fact, too, has been apparently independently rediscovered some years later and worked on steadily ever since (see for example [6]), expanding the number of optical elements and situations covered. It is important to remark, however, that the lens modeling approach in the later long ongoing series of papers view the multiplicative-constant phase term in the true form of the Fractional Fourier transform [as a problem or annoyance and usually omit it from consideration; this is odd as, for example, it is relatively simply to take the expression for lenses from [2] and repeat the development in [5] based on the simplified expression in [1] and exactly account for this multiplicative-constant phase term.
A principal aspect of this invention is the use of Fractional Fourier transform properties of lenses or other optical elements or environments to introduce one or more positive-definite optical transfer functions at various locations outside the Fourier plane so as to realize, or closely approximate, arbitrary non-positive-definite transfer functions. Specifically this aspect of the invention encompasses an optical system for realizing optical filtering through the use of at least one optical element outside the Fourier transform plane. By choice of the number of such elements, position of such elements, and the actual positive-define transfer function used for each element, arbitrary non-positive-definite transfer functions can be approximated by the entire system, and designs can be straightforwardly obtained by methods of approximation. However, there are several additional aspects to the invention relating to implementing, expanding, or utilizing this underlying aspect.
An intimately related aspect of the invention relates to realizations of the optical filtering effects of non-positive-definite optical transfer functions through the use of at least one positive-definite optical element outside the Fourier transform plane. The invention thus includes cases where only positive-definite optical elements are used to realize non-positive-definite optical transfer functions.
An additional aspect of the invention pertains to embodiment designs which can be straightforwardly obtained by methods of mathematical function approximation. An exemplary approximation method provided for in the invention leverages Hermite function expansions of the desired transfer function. This is advantageous in simplifying the approximation problem as the orthogonal Hermite functions diagonalize the Fourier transform and Fractional Fourier transform; the result is two-fold:
throughout the entire optical system the amplitude and phase affairs of each Hermite function are completely independent of those of the other Hermite functions
the Hermite function expansion of a desired transfer function will naturally have coefficients that eventually tend to zero, meaning that to obtain an arbitrary degree of approximation only a manageable number of Hermite functions need be handled explicitly.
Another aspect of the invention involves the determination of the position and/or amplitude distribution of a positive-definite optical element through use of the fractional Fourier transform.
Another aspect of the invention involves the determination of the position and/or amplitude distribution of a positive-definite optical element through use of Hermite function expansions, with or without approximations.
Another aspect of the invention relates to the use of at least one programmable light modulator is used as an optical element.
Another aspect of the invention relates to use of the invention to perform general optical computing functions, explicitly including those involving complex arithmetic.
Another aspect of the invention provides for realization of embodiments utilizing integrated optics, including monolithic or hybrid fabrication involving photolithography and/or beam-controlled fabrication methods.
Another aspect of the invention provides for multiple channel configurations which can be used, for example, in visual-color image processing or wide-spectrum image processing.
Another aspect of the invention provides for replacing one or more non-positive-definite filter elements in the principal and related aspects of the invention with one or more controllable optical phase shift elements, most generally in array form.
Another aspect of the invention uses the above variation (employing one or more controllable optical phase shift elements) to synthesize controllable lens and lens system functions such as controllable zoom and focus.
Another aspect of the invention provides for adapting the entire mathematical, method, and apparatus framework, so far built around the fractional Fourier transform operation and associated Hermite basis functions, to related mathematical transform operations associated with different basis functions that would arrive from non-quadratic graded-index media and/or environments.
The system and method herein can be applied to conventional lens-based optical image processing systems as well as to systems with other types of elements obeying Fractional Fourier optical models and as well to widely ranging environments such as integrated optics, optical computing systems, particle beam systems, radiation accelerators, and astronomical observation methods.
The incorporation of the method of the invention with classical, contemporary, and future methods of Fourier optics has significant synergistic value in attaining simple low-cost realizations of optical linear signal processing with non-positive-define transfer functions. Conventional methods for the creation of means for introducing positive-definite (amplitude variation without phase variation) optical transfer functions can be readily used in fabrication and conventional approximation methods can be used in transfer function design. Though use of Hermite function expansions, as used in one embodiment of the method, interaction of terms is minimized and only a manageable number of terms need be handled explicitly.